System for measuring multiphase flow using multiple pressure differentials

ABSTRACT

An improved method and system for measuring a multi-phase flow in a pressure flow meter. An extended throat venturi is used and pressure of the multi-phase flow is measured at three or more positions in the venturi, which define two or more pressure differentials in the flow conduit. The differential pressures are then used to calculate the mass flow of the gas phase, the total mass flow, and the liquid phase. The system for determining the mass flow of the high void fraction fluid flow and the gas flow includes taking into account a pressure drop experienced by the gas phase due to work performed by the gas phase in accelerating the liquid phase.

RELATED APPLICATION

This application is a continuation-in-part application of U.S. patentapplication Ser. No. 08/937,120 filed Sep. 24, 1997 abandoned.

CONTRACTUAL ORIGIN OF THE INVENTION

The United States Government has rights in this invention pursuant toContract No. DE-AC07-94ID13223 between the United States Department ofEnergy and Lockheed Martin Idaho Technologies Company.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to a flow meter for measuring the flow ofvery high void fraction multi-phase fluid streams. More particularly,the present invention relates to an apparatus and method in whichmultiple pressure differentials are used to determine mass flow rates ofgas and liquid phases of a predominantly gas fluid stream to therebydetermine the mass flow rate of each phase.

2. State of the Art

There are many situations where it is desirable to monitor multi-phasefluid streams prior to separation. For example, in oil well or gas wellmanagement, it is important to know the relative quantities of gas andliquid in a multi-phase fluid stream, to thereby enable determination ofthe amount of gas, etc. actually obtained. This is of criticalimportance in situations, such as off-shore drilling, in which it iscommon for the production lines of several different companies to betied into a common distribution line to carry the fuel back to shore. Inthe prior art, a common method for metering a gas is to separate out theliquid phase, but a separation system in not desirable for fiscalreasons. When multiple production lines feed into a common distributionline, it is important to know the flow rates from each production lineto thereby provide an accurate accounting for the production facilities.

In recent years, the metering of multi-phase fluid streams prior toseparation has achieved increased attention. Significant progress hasbeen made in the metering of multi-phase fluids by first homogenizingthe flow in a mixer then metering the pseudo single phase fluid in aventuri in concert with a gamma densitometer or similar device. Thisapproach relies on the successful creation of a homogenous mixture withequal phase velocities, which behaves as if it were a single phase fluidwith mixture density {overscore (ρ)}=αρ_(g)+(1−α)ρ_(l) where α is thevolume fraction of the gas phase, and ρ_(g) is the gas phase density andρ_(l) is the liquid phase density. This technique works well for flowswhich after homogenizing the continuous phase is a liquid phase. Whilethe upper limit of applicability of this approach is ill defined, it isgenerally agreed that for void fractions greater than about ninety toninety-five percent (90-95%) a homogenous mixture is very difficult tocreate or sustain. The characteristic unhomogenized flow in this voidfraction range is that of an annular or ring shaped flow configuration.The gas phase flows in the center of the channel and the liquid phaseadheres to and travels along the sidewall of the conduit as a thickfilm. Depending on the relative flow rates of each phase, significantamounts of the denser liquid phase may also become entrained in the gasphase and be conveyed as dispersed droplets. Nonetheless, a liquid filmis always present on the wall of the conduit. While the liquid generallyoccupies less than five percent (5%) of the cross-sectional volume ofthe flow channel, the mass flow rate of the liquid may be comparable toor even several times greater than that of the gas phase due to itsgreater density.

The fact that the phases are partially or fully separated, andconsequently have phase velocities which are significantly different(slip), complicates the metering problem. The presence of the liquidphase distorts the gas mass flow rate measurements and causesconventional meters, such as orifice plates or venturi meters, tooverestimate the flow rate of the gas phase. For example the gas massflow can be estimated using the standard equation$m_{g} = {\frac{{AC}_{c}Y}{\sqrt{1 - \beta^{4}}}\quad \sqrt{2\quad \rho_{g}\Delta \quad P}}$

where m_(g) is the gas mass flow rate, A is the area of the throat, ΔPis the measured pressure differential, ρ_(g) the gas density at flowconditions, C_(c) the discharge coefficient, and Y is the expansionfactor. In test samples using void fractions ranging from 0.997 to 0.95,the error in the measured gas mass flow rate ranges from 7% to 30%. Itis important to note that the presence of the liquid phase increases thepressure drop in the venturi and results in over-predicting the true gasmass flow rate. The pressure drop is caused by the interaction betweenthe gas and liquid phases. Liquid droplet acceleration by the gas,irreversible drag force work done by the gas phase in accelerating theliquid film and wall losses determine the magnitude of the observedpressure drop. In addition, the flow is complicated by the continuousentrainment of liquid into the gas, the redeposition of liquid from thegas into the liquid film along the venturi length, and also by thepresence of surface waves on the surface of the annular or ringed liquidphase film. The surface waves on the liquid create a roughened surfaceover which the gas must flow increasing the momentum loss due to theaddition of drag at the liquid/gas interface.

Other simple solutions have been proposed to solve the overestimation ofgas mass flow rate under multi-phase conditions. For example, Murdock,ignores any interaction (momentum exchange) between the gas and liquidphases and proposed to calculate the gas mass flow if the ratio of gasto liquid mass flow is known in advance. See Murdock, J. W. (1962). TwoPhase Flow Measurement with Orifices, ASME Journal of Basic Engineering,December, 419-433. Unfortunately this method still has up to a 20% errorrate or more.

Another example of a multi-phase measurement device in the prior art, isU.S. Pat. No. 5,461,930, to Farchi, et al, which appears to teach theuse of a water cut meter and a volumetric flow meter for measuring thegas and liquid phases. This invention is complicated must use a positivedisplacement device to measure the liquid and gas flow rates so it canavoid the problem of slip between the gas and liquid phases. This systemdoes not appear to be effective for liquid fractions below (5%-10%). Asmentioned earlier, other such prior art systems such as U.S. Pat. No.5,400,657 to Kolpak, et al, are only effective for multi-phase fluidflows where the gas fraction is 25% of the volume and the liquid is 75%of the volume.

Other volumetric measuring devices such as U.S. Pat. No. 4,231,262 toBoll, et al, measure a flow of solids in a gas stream. For example, coaldust in a nitrogen stream may be measured. Although these types ofdevices use pressure measuring structures, they are not able to addressthe problem of measuring a liquid fraction in a multi-phase flow wherethe liquid phase is less than 10% or even 5% of the overall volume.Measuring a liquid and gas is significantly different from measuring agas with a solid particulate. The mass of the liquid is significant andnot uniform throughout the gas. Incorrectly measuring the liquid throwsoff the overall measurements significantly. Furthermore, such deviceswhich have two pressure measuring points on the venturi throat, do nottake into account the fact that a pressure drop is caused by theinteraction between the gas and liquid phases and must be calculated foraccordingly.

While past attempts at metering multi-phase fluid streams have producedacceptable results below the ninety to ninety five percent (90-95%) voidfraction range, they have not provided satisfactory metering for thevery high void multi-phase flows which have less than five to ten(5-10%) non-gas phase by volume. When discussing large amounts ofnatural gas or other fuel, even a few percent difference in the amountof non-gas phase can mean substantial differences in the value of aproduction facility. For example, if there are two wells which produceequal amounts of natural gas per day. The first well produces, byvolume, 1% liquid and the second well produces 5% liquid. If aconventional mass flow rate meter is relied upon to determine the amountof gas produced, the second well will erroneously appear to produce asmuch as 20-30% more gas than the first well. Suppose further that theliquid produced is a light hydrocarbon liquid (e.g. a gas condensatesuch as butane or propane) which is valuable in addition to the naturalgas produced. Conventional meters will provide no information about theamount of liquid produced. Then if the amount of liquid produced isequally divided between the two wells, the value of the production fromthe first well will be overestimated while the production from thesecond well will be underestimated. To properly value the gas and liquidproduction from both wells, a method of more accurately determining themass flow rate of both the gas and liquid phases is required.

The prior art, however, has been incapable of accurately metering thevery high void multi-phase fluid streams. In light of the problems ofthe prior art, there is a need for an apparatus and method that is lesscomplex and provides increased accuracy for very high void multi-phasefluid streams. Such an apparatus and method should be physically rugged,simple to use, and less expensive than current technology.

SUMMARY OF THE INVENTION

It is an object of the present invention to provide an improvedapparatus and method for metering very high void multi-phase fluidstreams.

It is another object of the present invention to provide an apparatusand method which increases the accuracy of metering with respect to boththe gas phase and the liquid phase when measuring very high voidmulti-phase fluid streams.

It is still another object of the present invention to provide such anapparatus and method which does not require homogenization or separationof the multi-phase fluid in order to determine flow rate for each of thephases.

The above and other objects of the invention are realized in a specificapparatus for metering the phases of a multiple phase fluid. The flowmeter includes a cross-sectional area change in the flow conduit such asa venturi with an elongate passage. Disposed along the elongate passageis a converging section, an extended throat section, and a diffuser. Theflow meter also includes three or more pressure monitoring sites whichare used to monitor pressure changes which occur as the multi-phasefluid passes through the elongate passage and venturi. These pressurechanges, in turn, can be processed to provide information as to therespective flow rates of the phases of the multi-phase fluid. Bydetermining the flow rates of the components of the multi-phase fluid,the amount of natural gas, etc., can be accurately determined andaccounting improved.

BRIEF DESCRIPTION OF THE DRAWINGS

The above and other objects, features and advantages of the inventionwill become apparent from a consideration of the following detaileddescription presented in connection with the accompanying drawings inwhich:

FIG. 1 shows a side, cross-sectional view of a differential pressureflow meter with pressure measuring ports;

FIG. 2 shows a side, cross-sectional view of a differential pressureflow meter with an round shoulder;

FIG. 3 is a flow chart showing the steps required to calculate the massflow in a multiphase flow.

DETAILED DESCRIPTION

Reference will now be made to the drawings in which the various elementsof the present invention will be given numeral designations and in whichthe invention will be discussed so as to enable one skilled in the artto make and use the invention. It is to be understood that the followingdescription is only exemplary of the principles of the presentinvention, and should not be viewed as narrowing the pending claims.

Turning now to FIG. 1, there is shown another differential pressure flowmeter, generally indicated at 110. The differential pressure flow meter110 includes a venturi 114 formed by a sidewall 118 which defines afluid flow passage 122. The fluid flow passage 122 is segmented into aninlet section 126, a converging section 130, an extended throat section134, a diffuser section 138 and an outlet section 140.

The geometry and conduit diameter of the flow obstruction will varydepending on the particular application. The conduit may be larger orsmaller depending on the specific flow rate, pressure, temperature andother similar factors. One important characteristic of the flow meter isthat the preferred contraction ratio in the conduit should be between0.4 and 0.75. The contraction ratio is defined as the ratio of thethroat diameter 134 to the upstream conduit diameter 122. It is alsoimportant that the length of the throat is at least ten times thediameter of the throat. Of course, other throat lengths may be used.

An example of one possible set of conduit measurements will now begiven, but it should be realized that the actual geometry will depend onthe volume and size of the specific application. In one embodiment ofthe invention, the inlet section 126 has a diameter of about 3.8 cmadjacent the opening 142 at the upstream, proximal end 114 a of theventuri 114. The converging section 130 tapers inwardly from the inletsection 126 at an angle of about ten degrees (10°) until it connectswith the extended throat section 134, which has a diameter of about 2.5cm. The extended throat section 134 remains substantially the samediameter throughout its length and may be about 30 cm long to provideample length to determine acceleration differences between the variousphases. At the end of the extended throat section 134 b, the diffusersection 138 tapers outwardly at an angle of about three degrees (3°)until the diameter of the outlet section passage 140 is substantiallythe same as that at the inlet section 126 (i.e. 3 cm). It should berealized that many other specific geometric configurations could bedefined which have characteristics similar to the example above.

In order to monitor the pressure differentials caused by the changes influid velocity, the differential pressure flow meter shown in FIG. 1utilizes up to four different measurement points. Each pair of pressuremeasurement points defines a pressure differential. Only two pressuredifferential measurements are required to determine the gas and liquidflow rates. The preferred pressure differentials are ΔP₃ and ΔP₂.Pressure differential number three (ΔP₃) is defined as the pressurechange between points 150 and 154. Pressure differential number two(ΔP₂) is between points 154 and 158. The pressure differential ΔP₂ isimportant because it used for the calculation of the pressure dropexperienced by the gas phase due to the work performed by the gas phasein accelerating the liquid phase.

It should also be apparent based on this disclosure that the combinationof pressure differentials ΔP₃ and ΔP₀ or ΔP₂ and ΔP₀ may be usedinstead. Each of these combination work equally well, with the exceptionthat the numerical in the algorithm change. It is also important than anabsolute pressure and temperature measurement will be provided at theventuri inlet 142. Such a temperature measurement may be made, forexample, with a temperature sensor 157 which is in communication withthe multiphase flow and the flow processor 153.

Now the pressure ports will be described more specifically. A firstpressure measuring port 150 is disposed to measure the pressure in theinlet section 142. The first pressure measuring port 150 is connected toa pressure monitoring means, such as a pressure transducer 151, toprovide a pressure reading.

A second pressure measuring port 154 is provided at the entrance of theextended throat section 134. The second pressure measuring port 154 isdisposed adjacent the upstream, proximal end 134 a of the extendedthroat section 134. A pressure transducer 151 is also coupled to thesecond pressure measuring port 154.

Distally from the second pressure measuring port 154, but still withinthe extended throat section 134, is a third pressure monitoring port158. Preferably, the third pressure monitoring port 158 is disposedadjacent the distal end 134 b of the extended throat section 134, andadjacent the beginning 138 a of the diffuser section 138.

The respective pressure measuring ports 150, 154, and 158 are disposedin communication with a flow processor 153 or similar mechanism throughthe pressure monitoring means or pressure transducers 151, 155, and 159.The flow processor 153 enables the acquisition of the measured pressuredifferentials, and thus fluid flow rates in accordance with the presentinvention. Further, an accurate determination of the relativeacceleration of the two phases can also be obtained by comparing thepressure drop between the inlet section 126 (through measuring port 150)and the distal end 134 b of the extended throat section 134 (throughmeasuring port 158), as indicated at ΔP₀.

In an alternative embodiment of the invention, a fourth pressuremeasuring port 161 is disposed at the end of the extended throat 134 b.A fifth pressure measuring port 162 is disposed in the outlet section140 adjacent to the distal end 138 b of the diffuser section 138. Bothof these pressure measuring ports are coupled to pressure monitoringmeans or pressure transducer 163. The fourth and fifth monitoring portsallow a pressure differential ΔP₁ to be measured.

The pressure differential (ΔP₁) between the extended throat section 134and the distal end 138 b of the diffuser section 138 can also beanalyzed.

It should also be realized that different angles and lengths can be usedfor the venturi constriction and the extended throat of the venturitube. In fact, the converging section of the venturi is not required togradually taper. FIG. 2 shows a converging section 172 as formed by anannular shoulder in a venturi tube 170 to reduce the cross-sectionalarea of the inlet section. The preferred size of the radius of curvaturefor an annular shoulder 172 is about 0.652 cm. The converging sectioncan also be formed by placing a solid object in the conduit whichoccupies part but not all of the conduit cross-section.

It is vital that the correct method be used in the current invention toestimate the gas and fluid mass flow. Otherwise errors in the range of20% or more will be introduced into the measurements, as in the priorart. Reliable metering of high void fraction multi-phase flows over awide range of conditions (liquid loading, pressure, temperature, and gasand liquid composition) without prior knowledge of the liquid and gasmass flow rates requires a different approach than the simplemodification of the single phase meter readings as done in the priorart. Conceptually, the method of metering a fluid flow described here isto impose an acceleration or pressure drop on the flow field via astructure or venturi constriction and then observe the pressure responseof the device across two pressure differentials as described above.Because the multi-phase pressure response differs significantly fromthat of a single-phase fluid, the measured pressure differentials are aunique function of the mass flow rates of each phase.

As described above, the gas and liquid phases are strongly coupled. Whenthe gas phase accelerates in the converging section of the nozzle, thedenser liquid phase velocity appreciably lags that of the lighter gasphase. In the extended throat region, the liquid phase continues toaccelerate, ultimately approaching its equilibrium velocity with respectto the gas phase. Even at equilibrium, significant velocity differencesor slip will exist between the gas and liquid phases. A method foraccurately calculating the gas and liquid mass flows in an extendedventuri tube will now be described. (A derivation of the method is shownlater.) This method uses the four values which are determined thoughtesting. These values are: ΔP₃ which is the measured pressuredifferential across the venturi contraction, ΔP₂ which is the measuredpressure differential across the extended venturi throat, P which is theabsolute pressure upstream from the venturi (psi), and T which is thetemperature of the upstream flow. These measured values are used with anumber of predefined constants which will be defined as they are used.Alternatively, the pressure differentials ΔP₃ and ΔP₀, or the pressuredifferentials ΔP₀ and ΔP₂ may be used.

First, the gas density for the gas flow must be calculated based on thecurrent gas well pressure and temperature. This is done using thefollowing equation which uses English units. Any other consistent set ofunits may also be used with appropriate modifications to the equations.$\begin{matrix}{{rho}_{gw} = {{rho}_{g}\quad ( \frac{P + 14.7}{14.7} )\quad ( \frac{60 + 459.67}{T + 459.67} )}} & {{Equation}\quad 1}\end{matrix}$

where

rho_(g) is the density of natural gas (i.e. a mixture methane and otherhydrocarbon and non-hydrocarbon gases) at standard temperature (60° F.)and pressure (1 atmosphere) for a specific well;

P is the pressure upstream from the venturi in pounds per square inch(psi); and

T is the temperature upstream from the venturi in degrees Fahrenheit.

The value of rho_(g) will be different for various natural gascompositions and must be supplied by the well operator. At the standardtemperature (60° F.) and pressure (1 atmosphere) the value of rho_(g)for pure methane is 0.044 lb/ft³.

The second step is finding a normalized gas mass flow rate based on thesquare root of a pressure difference across the contraction multipliedby a first predetermined coefficient, and the square root of a measuredpressure differential across a venturi throat. The normalized gas massflow rate is found using the following equation: $\begin{matrix}{{mgm} = {A + {B\quad \sqrt{\Delta \quad P_{3}}} + {C\quad \sqrt{\Delta \quad P_{2}}}}} & {{Equation}\quad 2}\end{matrix}$

where

A, B, and C are experimentally determined constants required tocalculate gas mass flow rate;

ΔP₃ is the measured pressure differential across a venturi contraction;and

ΔP₂ is the measured pressure differential across a venturi throat. Thepreferred values for the constants in the equation above are as follows:A is −0.0018104, B is 0.008104 and C is −0.0026832 when pressure is inpounds per square inch (psi), density in lbs/ft³ and mass flow rate inthousands of mass lbs/minute. Of course, these numbers are determinedexperimentally and may change depending on the geometry of the venturi,the fluids used, and the system of units used.

Calculating the normalized gas mass flow rate is important because itallows the meter to be applied to the wells or situations where thepressure or meter diameter for the liquids present are different thanthe conditions under which the meter was originally calibrated. Thismeans that the meter does not need to be calibrated under conditionsidentical to those present in a particular application and that themeter may be sized to match the production rate from a particular well.

The functional form of Equation 2 is arrived at by derivation from theconservation of mass and energy followed by a simplifying approximation.Other functional forms of Equation 2 can be used with equivalentresults. The functional form of Equation 2 is consistent with theconservation laws and provides a good representation of the calibrationdata.

The third step is computing a gas mass flow rate using the normalizedgas mass flow rate, the gas density, and a contraction ratio of theventuri tube. The equation for calculating the gas mass flow rate fromthese quantities is $\begin{matrix}{{mg} = {{mgm} \cdot A_{t} \cdot \frac{\sqrt{{rho}_{gw}}}{\sqrt{1 - \beta^{4}}}}} & {{Equation}\quad 3}\end{matrix}$

where

mgm is the normalized gas mass flow rate;

A_(t) is the venturi throat area;

β is the contraction ratio of the throat area; and

rho_(gw) is the gas density at current well conditions.

The fourth step is estimating the gas velocity in the venturi tubethroat. The equation for estimating the gas velocity is: $\begin{matrix}{u_{g}\frac{m_{g}}{{rho}_{g} \cdot A_{t}}} & {{Equation}\quad 4}\end{matrix}$

where

m_(g) is the gas mass flow rate;

rho_(g) is the density of the gas phase for a specific well; and

A_(t) is the venturi throat area.

The fifth step is calculating the pressure drop experienced by the gasphase due to work performed by the gas phase in accelerating the liquidphase between an upstream pressure measuring point and a pressuremeasuring point in the distal end of the venturi throat. The pressuredrop is calculated as follows: $\begin{matrix}{{\Delta \quad P_{gl3}} = {{\Delta \quad P_{3}} - {\frac{1}{2} \cdot {rho}_{gw} \cdot u_{g}^{2} \cdot ( {1 - \beta^{4}} )}}} & {{Equation}\quad 5}\end{matrix}$

where

ΔP₃ is the measured pressure differential across a venturi contraction;

rho_(gw) is gas density at well conditions;

u_(g) is the gas velocity in the venturi throat; and

β is the contraction ratio of the throat area to the upstream area.

It is important to note that the calculations outlined in steps two andfive are important because they allow for estimating the mass flow ofeach phase.

Step six is estimating the liquid velocity (u_(l)) in the venturi throatusing the calculated pressure drop experienced by the gas phase due towork performed by the gas phase. This is performed as follows$\begin{matrix}{u_{t} = \sqrt{\frac{2\quad ( {{\Delta \quad P_{3}} - {\Delta \quad P_{gl3}}} )}{{rho}_{l} \cdot \lbrack {( {1 + \beta^{4}} ) + {gcfw}} \rbrack}}} & {{Equation}\quad 6}\end{matrix}$

where

ΔP₃ is the measured pressure differential across a venturi contraction;

ΔP_(gl3) is the pressure drop experienced by the gas-phase due to workperformed by the gas phase on the liquid phase;

rho_(l) is the liquid density; and

gcfw is a constant which characterizes wall friction. The preferredvalue for gcfw is defined as 0.062. This value may be adjusted dependingon different venturi geometries or different fluids.

The seventh step is computing the friction between the liquid phase anda wall in the venturi which is performed: $\begin{matrix}{f = {{gcfw} \cdot \frac{1}{2} \cdot {rho}_{l} \cdot u_{l}^{2}}} & {{Equation}\quad 7}\end{matrix}$

where

gcfw is a constant which characterizes wall friction;

rho_(l) is the liquid density; and

u_(l) is the liquid velocity in the venturi throat.

The eighth step is calculating the total mass flow rate based on themeasured pressure in the venturi throat, the calculated friction and thegas velocity. The equation for this is: $\begin{matrix}{m_{t} = {\frac{2\quad ( {{\Delta \quad P_{3}} - f} )}{( {1 - \beta^{4}} ) \cdot u_{g}} \cdot A_{t}}} & {{Equation}\quad 8}\end{matrix}$

where

ΔP₃ is the measured pressure differential across a venturi contraction;

β is the contraction ratio of the throat diameter to the upstreamdiameter; and

u_(g) is the gas velocity in the venturi throat.

The liquid mass flow rate can now be calculated as the differencebetween the total and gas mass flow rates.

m _(l)=(m _(t) −m _(g))  Equation 9

wherein

m_(t) is the total mass flow rate; and

m_(g) is the gas mass flow rate.

Calculating the gas mass flow rate, total mass flow rate, and liquidmass flow rate using the method outlined above is much more accuratethan the prior art. The accuracy of method outlined above is within ±4%for the gas phase, ±5% for the liquid phase, and ±4% for the total massflow. This accuracy can even be increased using measured calibrationsfor a specific installation to benchmark the readings.

FIG. 3 shows a summary of the method used to accurately calculate themass flow through the elongated venturi. The method for determining themass flow of the high void fraction fluid flow and the gas flow includessteps which were described with Equations 1-9. Referring to FIG. 3, thefirst step is calculating a gas density for the gas flow 210. The nexttwo steps are finding a normalized gas mass flow rate through theventuri 220 and computing a gas mass flow rate 230. The following stepis estimating the gas velocity in the venturi tube throat 240. The nextstep is calculating the pressure drop experienced by the gas-phase dueto work performed by the gas phase in accelerating the liquid phasebetween the upstream pressure measuring point and the pressure measuringpoint in the venturi throat 250. Yet another step is estimating theliquid velocity 260 in the venturi throat using the calculated pressuredrop experienced by the gas-phase due to work performed by the gasphase. Then the friction is computed 270 between the liquid phase and awall in the venturi tube. Finally, the total mass flow rate based onmeasured pressure in the venturi throat is calculated 280 and the liquidmass flow rate is determined 290.

Theoretical Gas Mass Flow Rate

Now a discussion of the theoretical derivations will be outlined whichproduced the method described above. The theoretical derivation is basedon the physical laws describing the conservation of mass and energy forboth the gas and liquid phases. The conservation of mass and energyequations for each phase are shown below where the subscript 1 denotesthe upstream condition measured at 142 by pressure tap 150 in FIG. 1,and the subscript 2 denotes the venturi throat entrance measured at 134a by pressure tap 154. ΔP_(gl3) is the pressure drop experienced by thegas phase due to work done by the gas phase in accelerating the liquidphase between the pressure measuring location at the beginning of theelongated throat and the pressure measuring location at the end of thethroat. It is assumed that only the liquid phase is in contact with thewall, f_(w) is the wall friction coefficient and G_(c) is a geometryfactor which accounts for the acceleration of the fluid in the venturicontraction and the surface area of the contraction. $\begin{matrix}{m_{g} = {{\alpha_{1}\rho_{g}u_{g1}A_{1}} = {\alpha_{2}\rho_{g}u_{g2}A_{2}}}} & {{Equations}\quad 10} \\{m_{l} = {{( {1 - \alpha_{1}} )\rho_{l}u_{l1}A_{1}} = {( {1 - \alpha_{2}} )\rho_{l}u_{l2}A_{2}}}} & \quad \\{{P_{1} + {\frac{1}{2}\rho_{g}u_{g1}^{2}}} = {P_{2} + {\frac{1}{2}\rho_{g}u_{g2}^{2}} + {\Delta \quad P_{gl3}}}} & \quad \\{{P_{1} + {\frac{1}{2}\rho_{l}u_{l1}^{2}}} = {P_{2} + {\frac{1}{2}\rho_{l}u_{l2}^{2}} - {\Delta \quad P_{gl3}} + {G_{c}f_{w}\frac{1}{2}\rho_{l}u_{l2}^{2}}}} & \quad\end{matrix}$

In Equations 10, α is void fraction, ρ_(g) is density of a gas atstandard temperature, u_(g) is the gas velocity, A₁ is the conduit areaupstream of the venturi, A₂ is the conduit area in the venturi throat,and P₁ and P₂ are the pressures at locations 142 (tap 150) and 134 a(tap 154) in the conduit.

The gas phase energy equation can be rewritten using the equation forthe gas phase mass flow rate, where D is the diameter of the upstreampiping, d is the throat diameter, β=d/D is the contraction ratio, andΔP₃=P₂−P₁ is the pressure drop across the contraction. $\begin{matrix}{{\Delta \quad P_{3}} = {{\frac{1}{2}\frac{m_{g}^{2}}{\rho_{g}\alpha_{2}^{2}A_{2}^{2}}( {1 - {( \frac{\alpha_{2}}{\alpha_{1}} )^{2}\beta^{4}}} )} + {\Delta \quad P_{gl3}}}} & {{Equation}\quad 11}\end{matrix}$

With the approximation that α₁ and α₂≅1, the modified orifice equationresults. $\begin{matrix}{{\Delta \quad P_{3}} \approx {{\frac{1}{2}\frac{m_{g}^{2}}{\rho_{g}A^{2}}( {1 - \beta^{4}} )} + {\Delta \quad P_{gl3}}}} & {{Equation}\quad 12}\end{matrix}$

For single-phase flow ΔP_(gl3) is equal to zero and the equation issolved directly for the mass flow rate m_(g). In practice, thesingle-phase result is modified by the addition of an empirical constantC_(c) which accounts for the true discharge characteristics (non-idealone-dimensional behavior and friction losses) of the nozzle and Y whichtakes compressibility effects into account. $\begin{matrix}{m_{{gl}\quad \varphi} = {\frac{C_{c}{AY}}{\sqrt{1 - \beta^{4}}}\sqrt{2\rho_{g}\Delta \quad P_{3}}}} & {{Equation}\quad 13}\end{matrix}$

As shown in the introduction, if the Equation 13 above is used undermultiphase conditions, the mass flow rate of the gas phase can besignificantly overestimated. Under multiphase conditions the mass flowrate of the gas phase is given by: $\begin{matrix}{m_{g} = {\frac{C_{2\quad \varphi}\alpha_{2}A_{2}Y}{\sqrt{1 - {( \frac{\alpha_{2}}{\alpha_{1}} )^{2}\beta^{4}}}}\sqrt{2{\rho_{g}( {{\Delta \quad P_{3}} - {\Delta \quad P_{gl3}}} )}}}} & {{Equation}\quad 14}\end{matrix}$

where α₂A₂ represents the cross sectional area occupied by the gasphase. When ΔP₃ is large with respect to ΔP_(gl3) the quantity under theradical can be approximated by

{square root over (ΔP ₃ −ΔP _(gl3))}  Equation 15

≈{square root over (ΔP ₃)}−C _(gl3)

×{square root over (ΔP _(gl3))}

where C_(gl3) is a constant that is determined experimentally.Empirically it has been found that ΔP_(gl3) can be replaced by afunction of ΔP₂, the pressure drop in the extended throat, withappropriate choice of constants. The mass flow rate of gas under bothsingle phase and multiphase conditions now becomes $\begin{matrix}{{m_{g}\frac{C_{2\quad \varphi}{AY}}{\sqrt{1 - \beta^{4}}}{\sqrt{2\rho_{g}}\lbrack {\sqrt{\Delta \quad P_{3}} - {C_{2} \times \sqrt{P_{2}}}} \rbrack}},} & {{Equation}\quad 16}\end{matrix}$

where it has been assumed that α₂≈α₁≈1. The constants C_(2φ) and C₂ havebeen determined empirically and the validity of the equation has beentested over a wide range of conditions. It is important to note thatthis method can be used not only with natural gas production but othergas and liquid phase compositions. In addition, it is also important torecognize that Equations 10-16 are used to derive calculation steps inthe calculation method.

We have assumed that α₂≈α₁≈1, making Equation 16 above only approximate.The statistical fitting procedure used to determine the constants C_(2φ)and C₂ implicitly determines a weighted mean value of α. Because α doesnot appear explicitly and is unknown, there is an uncertainty of ±1-2%over the void fraction range 0.95<α<1.0, implicit in the equation. If αor (1−α) is independently measured, the observed measurementuncertainties can be significantly reduced. The uncertainty can also besignificantly reduced if, at installation, the actual flow rates areaccurately known. If this measurement is available then the meterreading can be adjusted to reflect the true value and the uncertainty inthe gas phase mass flow rate measurement can be reduced to less than0.5% of reading if the gas and liquid flow rates change by less than 50%or so over time. The repeatability of the measurement is essentially therandom uncertainty in the pressure measurements, less than about 0.5% ofreading.

Total and Liquid Mass Flow Rate

If the ratio of liquid to gas flow rate is known a priori with certaintythen the mass flow rate of the liquid phase can be directly obtainedfrom m_(l)=m_(g)(m_(l)/m_(g))_(known). Note that because the liquid massflow rate is only a fraction (0-30%) of the gas mass flow rate theuncertainty in the measurement is magnified. For instance, ifm_(l)/m_(g)=0.01, a 1% error in m_(g) is magnified to become a 100% ofreading error for the liquid phase. An additional fixed error of 1% inthe ratio m_(l)/m_(g) results in a 200% of reading total error for theliquid phase. This approach, of course, assumes that the m_(l)/m_(g)ratio remains constant over time.

Unfortunately, without accurate independent knowledge of α or (1−α) theliquid mass flow rate cannot be obtained directly from one-dimensionaltheory. The velocity of the liquid phase can, however, be estimateddirectly as now described. Once the mass flow rate of the gas phase isdetermined the ΔP_(gl3) term can be estimated from the gas phase energyequation: $\begin{matrix}{{\Delta \quad P_{gl3}} \approx {{\Delta \quad P_{3}} - {\frac{1}{2}\frac{m_{g}^{2}}{\rho_{g}A^{2}}( {1 - \beta^{4}} )}}} & {{Equation}\quad 17}\end{matrix}$

Equation 17 allows us to derive Equation 5 in the calculation method.Rearranging the liquid phase energy equation yields $\begin{matrix}{{{\Delta \quad P_{3}} + {\Delta \quad P_{gl3}}} = {{\frac{1}{2}\quad \rho_{l}{u_{l2}^{2}( {1 - \frac{u_{l1}^{2}}{u_{l2}^{2}}} )}} + {G_{c}f_{w}\quad \frac{1}{2}\quad \rho_{l}u_{l2}^{2}}}} & {{Equation}\quad 18}\end{matrix}$

and using the expression for the mass flow rate of liquid results in:$\begin{matrix}{{{\Delta \quad P_{3}} + {\Delta \quad P_{gl3}}} = {{\frac{1}{2}\quad \rho_{l}{u_{l2}^{2}( {1 - {\frac{( {1 - \alpha_{2}} )^{2}}{( {1 - \alpha_{1}} )^{2}}\quad \beta^{4}}} )}} + {G_{c}f_{w}\quad \frac{1}{2}\quad \rho_{l}u_{l2}^{2}}}} & {{Equation}\quad 19}\end{matrix}$

With the assumption that$\frac{( {1 - \alpha_{2}} )^{2}}{( {1 - \alpha_{1}} )^{2}}\quad \beta^{4}{\operatorname{<<}1}$

the liquid velocity u_(l2) can be estimated. If (1−α) is known then theliquid mass flow rate could be estimated directly from m_(l)=(1−α₂)ρu_(l2)A. Unfortunately, (1−α) cannot be accurately estimated directlyfrom the differential pressure data; it must be independently measuredto pursue this approach.

If we consider the gas and liquid phases together but allow theirvelocities to differ, the total mass flow rate can be written as:$\begin{matrix}{m_{t} = {{m_{g} + m_{l}} = {( {{\alpha \quad \rho_{g}} + {\frac{( {1 - \alpha} )}{S}\quad \rho_{l}}} )u_{g}A}}} & {{Equation}\quad 20}\end{matrix}$

where the density term in brackets is the effective density, ρ_(slip)and S=u_(g)/u_(l) which is ratio of the gas velocity to the liquidvelocity or slip. Since m_(t) is constant throughout the venturi, itallows us to write the pressure drop ΔP₃ as $\begin{matrix}{{\Delta \quad P_{3}} = {{\frac{1}{2}( {{\alpha \quad \rho_{g}} + {\frac{( {1 - \alpha} )}{S}\rho_{l}}} )u_{g}^{2}\quad ( {1 - \beta^{4}} )} + {G_{c}f_{w}\frac{1}{2}\rho_{l}u_{l2}^{2}}}} & {{Equation}\quad 21}\end{matrix}$

The second term on the right hand side is the friction loss assumingthat only the liquid phase is in contact with the wall. The equation canbe rearranged to yield the total mass flow rate $\begin{matrix}{m_{t} = {{( {{\alpha \quad \rho_{g}} + {\frac{( {1 - \alpha} )}{S}\rho_{l}}} )u_{g}A} = \frac{2( {{\Delta \quad P_{3}} - {G_{c}f_{w}\frac{1}{2}\rho_{l}u_{l2}^{2}}} )A}{( {1 - \beta^{4}} ) \cdot u_{g}}}} & {{Equation}\quad 22}\end{matrix}$

The total mass flow rate m_(t) can then be obtained directly from ΔP₃once u_(g) is estimated from the measured value of m_(g),u_(g)=m_(g)/ρ_(g)A and the liquid velocity is calculated by solvingequation 19 for u_(l2). The total mass flow rate using this method is ameasurement with an uncertainty of ±4% of the actual measured flow. Inprinciple, (since the total mass flow rate is the sum of the gas andliquid mass flow rates) the liquid mass flow rate can now be obtaineddirectly from m_(l)=m_(t)−m_(g). The liquid mass flow rate can then beobtained within ±5% of the total mass flow rate.

As previously noted in the discussion of the measurement of the gas massflow rate, if the flow rates of each phase are accurately known at thetime of installation, measurement performance over a reasonable range ofmass flow rates can be significantly enhanced. The uncertainty in thegas mass flow rate measurement can be reduced to <0.5% of reading bybenchmarking even if the gas and/or liquid mass flow rates change by±50%. Similarly, the uncertainty in the total mass flow rate can bereduced by <2% of reading for the same ±50% changes in gas and/or liquidmass flow rates. The corresponding improvement in accuracy of the liquidphase measurement is also significant. Because the liquid mass flow ratemeasurement is dependent on both the gas phase and total mass flow ratemeasurements, the uncertainty is also sensitive to changes in both gasand liquid mass flow rate. If the liquid mass flow rate measurement isbenchmarked at an initial value, the data indicate that the accuracyattainable is ±20% of reading for changes in gas mass flow rate in therange of ≦±15% and/or changes in liquid mass flow rate in the range of≦±25%. The uncertainty in the liquid mass flow rate quoted in terms ofpercent of total mass flow rate becomes ±1%.

Measurement uncertainties can be significantly reduced if flow rates areaccurately known at time of meter installation or periodically measuredby separation and separate metering during the service life of the meterand the well. Because the liquid phase is generally only a smallfraction of the total mass flow rate the uncertainty in its measurementis inherently high. If the void fraction a is accurately andindependently measured, the liquid mass flow rate can be calculateddirectly from m_(l)−(1−α)l_(l)u_(l2)A where the u_(l2) the liquidvelocity is obtained as described above from equation 19. The voidfraction may be accurately and independently measured using a gamma rayattenuation densitometer or through ultrasonic film thicknessmeasurements. This approach has been shown to significantly reduce theuncertainty in the liquid mass flow rate measurement.

I claim:
 1. A method for determining mass flow rates of gas and liquidphases in a flow, the method comprising: providing a venturi having aninlet, an outlet, and an extended throat disposed between the venturiinlet and venturi outlet; passing a high void fraction liquid and gasflow through the venturi to create a first pressure differential betweenthe venturi inlet and the extended throat, and a second pressuredifferential within the extended throat; measuring the first and secondpressure differentials; processing the first and second pressuredifferentials, including the first pressure differential between apressure measuring point in the throat inlet and a pressure measuringpoint in the throat outlet, and the second pressure differential betweena pressure measuring point in the venturi inlet and a measuring point inthe throat outlet; determining a pressure drop in the gas phase due towork performed by the gas phase in accelerating the liquid phase; anddetermining the respective mass flow rates of gas and liquid of themulti-phase flow.
 2. The method according to claim 1, wherein the methodfurther comprises, estimating the friction of the liquid phase on theconduit wall from the measured pressure differentials and the pressuredrop calculated for the gas phase due to work performed by the gas phasein accelerating the liquid phase, and further using this value tocalculate the sum of the mass flow rates of the gas and liquid phases.3. A system for facilitating measurement of respective mass flow ratesof gas and liquid phases of a high void fraction multiphase flow, thesystem comprising: a venturi comprising: an inlet section ofpredetermined diameter and having first and second ends; a convergingsection having first and second ends, said first end of said convergingsection being attached to said second end of said inlet section; anextended throat having first and second ends, said first end of saidextended throat being attached to said second end of said convergingsection; a diffuser section having first and second ends, said first endof said diffuser section being attached to said second end of saidextended throat; an outlet section of predetermined diameter and havingfirst and second ends, said first end of said outlet section beingattached to said second end of said diffuser section; a plurality ofpressure measuring points including: a first pressure measuring pointlocated proximate said inlet section; a second pressure measuring pointlocated proximate said second end of said converging section; and athird pressure measuring point located proximate said second end of saidextended throat; and a processor operatively coupled with the pluralityof pressure measuring points, the processor being configured tocalculate a pressure drop experienced by the gas phase of the high voidfraction multiphase flow due to work performed by the gas phase inaccelerating the liquid phase through the venturi based at leastpartially upon measurements taken at the plurality of pressure measuringpoints.
 4. The venturi as recited in claim 3, wherein said convergingsection defines a flow path tapered from said first end of saidconverging end to said second end of said converging section.
 5. Theventuri as recited in claim 3, wherein said converging section includesan annular shoulder that defines a flow path having two differentdiameters.
 6. The venturi as recited in claim 3, wherein said convergingsection includes a structure partly obstructing said converging sectionso as to define a flow path of at least two different cross-sectionalareas.
 7. The venturi as recited in claim 3, further comprising fourthand fifth pressure measuring points located, respectively, proximatesaid first end of said diffuser section, and proximate said outletsection.
 8. The venturi as recited in claim 3, wherein the venturidefines a contraction ratio in the range of about 0.4 to about 0.75. 9.The venturi as recited in claim 3, wherein said extended throat has alength-to-diameter ratio of at least about ten (10).
 10. A differentialpressure flow meter suitable for use in facilitating measurement ofrespective mass flow rates of gas and liquid phases of a multiphaseflow, comprising: a venturi including: an inlet section of predetermineddiameter and having first and second ends; a converging section havingfirst and second ends, said first end of said converging section beingattached to said second end of said inlet section; an extended throathaving first and second ends, said first end of said extended throatbeing attached to said second end of said converging section; a diffusersection having first and second ends, said first end of said diffusersection being attached to said second end of said extended throat; anoutlet section of predetermined diameter and having first and secondends, said first end of said outlet section being attached to saidsecond end of said diffuser section; and a plurality of pressuremeasuring points including: a first pressure measuring point locatedproximate said inlet section; a second pressure measuring point locatedproximate said second end of said converging section; and a thirdpressure measuring point located proximate said second end of saidextended throat; a first pressure transducer connected to said first andsecond pressure measuring points, a second pressure transducer connectedto said second and third pressure measuring points, and a third pressuretransducer connected to said first and third pressure measuring points;and a flow processor in communication with said first, second, and thirdpressure transducers configured to calculate a normalized gas flow ratebased at least partially on a plurality of measurements taken from theplurality of pressure measuring points.
 11. The differential pressureflow meter as recited in claim 10, further comprising a temperaturesensor in communication with both the multiphase flow and the flowprocessor.
 12. The differential pressure flow meter as recited in claim10, further comprising a fourth pressure transducer in communicationwith the multiphase flow at a first location proximate said second endof said extended throat and at a second location proximate said outletsection.
 13. The differential pressure flow meter as recited in claim10, wherein said venturi defines a contraction ratio in the range ofabout 0.4 to about 0.75.
 14. The differential pressure flow meter asrecited in claim 10, wherein said extended throat has alength-to-diameter ratio of at least about ten (10).
 15. Thedifferential pressure flow meter as recited in claim 10, furthercomprising a gamma ray attenuation densitometer configured forcommunication with the multiphase flow.
 16. The differential pressureflow meter as recited in claim 10, further comprising an ultrasonic filmthickness measurement device configured for communication with themultiphase flow.
 17. In conjunction with a venturi having an inlet andoutlet section, the inlet section being in communication with aconverging section and the outlet section being in communication with adiffuser section, and an extended throat being disposed between theconverging section and the diffuser section, a method for determiningrespective mass flow rates of gas and liquid phases of a multiphase flowof known flow rate passing through the venturi, the method comprising:determining a gas phase density; determining a normalized gas phase massflow rate through the venturi using a first pressure differentialmeasured between a point proximate the inlet and proximate the outlet ofthe extended throat section and a second pressure differential measuredbetween a point proximate the inlet and proximate the outlet of theconverging section of the venturi and; determining an actual gas phasemass flow rate using the gas phase density and the normalized mass flowrate of the gas phase; and determining a liquid phase mass flow rate bysubtracting the actual gas phase mass flow rate from the known flow rateof the multiphase flow.
 18. The method as recited in claim 17 whereindetermining a density for the gas phase comprises calculating said gasphase density (rho_(gw)) using the equation:${rho}_{gw} = {{rho}_{g}\quad ( \frac{P + 14.7}{14.7} )\quad ( \frac{60 + 459.67}{T + 459.67} )}$

wherein, rho_(g) is a natural gas density, at a temperature of aboutsixty (60) degrees F. and a pressure of about one (1) atmosphere,associated with a particular multiphase flow source; P is a pressure ofthe multiphase flow upstream of the venturi; and T is a temperature ofthe multiphase flow upstream of the venturi.
 19. The method as recitedin claim 17 wherein determining a normalized gas phase mass flow ratecomprises calculating said normalized gas mass flow rate (mgm) using theequation: mgm=A+B{square root over (Δ)}P ₃ +C{square root over (Δ)}P ₂wherein, A, B, and C are experimentally determined constants required tocalculate said normalized gas phase mass flow rate; ΔP₂ comprises saidfirst pressure differential; and ΔP₃ comprises said second pressuredifferential.
 20. The method as recited in claim 19 further comprisingselecting said constant A to be equal to −0.0018104, B to be equal to0.008104 and C to be equal to −0.0026832, pressure in pounds per squareinch (psi), mass flow rate in thousands of mass lbs/minute, density inpounds per cubic foot, and area in square inches.
 21. The method asrecited in claim 17 wherein determining said actual gas phase mass flowrate comprises calculating said actual gas phase mass flow rate (mg)using the equation:$m_{g} = {{mgm} \cdot A_{t} \cdot \frac{\sqrt{{rho}_{gw}}}{\sqrt{1 - \beta^{4}}}}$

wherein, mgm is the normalized gas phase mass flow rate; A_(t) is anarea of the extended throat in square inches; β is a contraction ratioof the throat area to an upstream area; and rho_(gw) is said gas phasedensity.
 22. The method as recited in claim 17 further comprisingmeasuring said first pressure differential between two predeterminedpoints proximate the extended throat.
 23. The method as recited in claim17 further comprising measuring a pressure differential between a firstpoint located upstream of the converging section and a second pointlocated downstream of the converging section.
 24. In a device having ageometry that defines a flow path, a method suitable for determiningrespective mass flow rates of gas and liquid phases of a multiphase flowpassing through the flow path defined by the device, the methodcomprising: accelerating the multiphase flow; determining a response ofthe multiphase flow to said acceleration, said response being manifestedas at least one pressure change of the multiphase flow; determining agas phase density; determining a normalized gas phase mass flow ratethrough the flow path based upon said response of the multiphase flow tosaid acceleration; determining an actual gas phase mass flow rate basedupon said gas phase density, said normalized mass flow rate of the gasphase, and the geometry of the device; determining a gas phase velocitybased upon said gas phase mass flow rate and the geometry of the device;determining a pressure drop experienced by the gas phase due to workperformed by the gas phase in accelerating the liquid phase betweenpredetermined points in the flow path defined by the device, saidpressure drop being determined based upon said response of themultiphase flow to said acceleration, said gas phase velocity, said gasphase density, and the geometry of the device; determining a liquidphase velocity at a selected location in the flow path based upon saiddetermined pressure drop, the geometry of the device, a frictionconstant, a liquid phase density, and said response of the multiphaseflow to said acceleration; determining a friction value between theliquid phase and the device using said liquid phase velocity and saidliquid phase density; and determining a total mass flow rate of themultiphase flow based upon the geometry of the device, said response ofthe multiphase flow to said acceleration, said friction value, and saidgas phase velocity.
 25. The method as recited in claim 24, furthercomprising determining a liquid phase mass flow rate by subtracting saidactual gas phase mass flow rate from said total mass flow rate of themultiphase flow.
 26. The method as recited in claim 24, whereindetermining a response of the multiphase flow to said accelerationcomprises measuring at least two pressure differentials of themultiphase flow in the device.
 27. The method as recited in claim 24,wherein determining a gas phase density (rho_(gw)) comprises calculatingsaid gas phase density using the equation:${rho}_{gw} = {{rho}_{g}\quad ( \frac{P + 14.7}{14.7} )\quad ( \frac{60 + 459.67}{T + 459.67} )}$

wherein, rho_(g) is a natural gas density, at a temperature of aboutsixty (60) degrees F. and a pressure of about one (1) atmosphere,associated with a particular multiphase flow source; P is a pressure ofthe multiphase flow upstream of the device; and T is a temperature ofthe multiphase flow upstream of the device.
 28. The method as recited inclaim 24, wherein determining a normalized gas phase mass (mgm) flowrate comprises calculating said normalized gas phase mass flow rateusing the following equation: mgm=A+B{square root over (Δ)}P ₃ +C{squareroot over (Δ)}P ₂ wherein, A, B, and C are expimentally determinedconstants required to calculate said gas phase mass flow rate; ΔP₃ is apressure differential measured across a first set of predeterminedpoints in the device; and ΔP₂ is a pressure differential measured acrossa second set of predetermined points in the device.
 29. The method asrecited in claim 28 wherein said constant A is equal to −0.0026832, B isequal to 0.008104 and C is equal to −0.0026832, pressure is in poundsper square inch (psi), mass flow rate is in thousands of masslbs/minute, and density is in pounds per cubic foot, and area is insquare inches.
 30. The method as recited in claim 24 wherein determininga friction value (f) between the liquid phase and the device calculatingsaid friction value using the following equation:$f = {{gcfw} \cdot \frac{1}{2} \cdot {rho}_{1} \cdot {uu}_{1}^{2}}$

wherein, gcfw is a constant which characterizes friction between thedevice and the multiphase flow; rho_(l)is said liquid phase density; andu_(l) is said liquid phase velocity.
 31. The method as recited in claim30 further comprising selecting gcfw to have a value of about 0.062. 32.In a device having a geometry that defines a flow path, a methodsuitable for determining respective mass flow rates of gas and liquidphases of a multiphase flow of known flow rate passing through the flowpath defined by the device, the method comprising: accelerating themultiphase flow; determining a response of the multiphase flow to saidacceleration, said response being at least partially measured by atleast one pressure differential between the inlet and outlet of aconstant area region of the flow path defined by the device; determininga gas phase density; determining a normalized gas phase mass flow ratethrough the flow path based upon said response of the multiphase flow tosaid acceleration; determining an actual gas phase mass flow rate basedupon said gas phase density, said normalized mass flow rate of the gasphase, and the geometry of the device; and determining a liquid phasemass flow rate by subtracting said actual gas phase mass flow rate fromthe known flow rate of the multiphase flow.
 33. The method as recited inclaim 32, wherein determining a response of the multiphase flow to saidacceleration comprises measuring at least two pressure differentials ofthe multiphase flow in the device.